1. Field of the Disclosure
The present disclosure relates generally to constructing a polar code, and more particularly, to constructing a polar code for an arbitrary class of channels, which is universally good for the class of channels with a certain Frame Error Rate (FER) across all of the channels.
2. Description of the Related Art
Polar codes are the first and currently only family of codes with explicit construction (i.e., no ensemble to pick from) and low-complexity encoding and decoding algorithms that achieve the capacity over a certain class of channels. A polar transformation is defined as a multiplication of an input vector by a polarization matrix
      G          ⊗      l        =                    [                                            1                                      0                                                          1                                      1                                      ]                    ⊗        l              .  
Polar code construction, or channel polarization, is based on the observation that as a length n=2l of a polar transformation increases, the observed bit-channels at the input are polarized so that they become either noiseless (perfect) channels or completely noisy channels. A polar code is constructed by transmitting information bits over the noiseless channels, also referred to as good bit-channels, while restricting (or freezing) the input to the noisy channels, also referred to as bad bit-channels, to zeros.
Constructing polar codes (i.e., finding good bit-channels) is, in general, a difficult problem. There are some heuristic and approximate algorithms that attempt to solve the problem. However, they only concern one given channel and do not disclose a method of constructing a polar code that is universally good for an arbitrary class of channels.
Another complicating factor is that the construction of polar codes, in general, depends on the characteristics of the underlying channel. As a result, if a polar code is optimized for transmission across a certain channel, it may not be good for transmission over another channel. This is a challenge in constructing polar codes suitable for a practical application, because in a communications system, the underlying channel varies. Thus, there is a need for an apparatus and method of constructing a polar code that is robust to channel variations.